We study the relation of Markov fluid queues and QBD processes in this paper. Ahn and Ramaswami presented results about this relation and provided a stochastic interpretation based reasoning in [1]. In the current work, first we provide an algebraic proof for that relation. After that, we present a negative result about the potential extension of the QBD based analysis Markov fluid queues to Markov fluid queues with two buffers. We present a 2-dimensional QBD process, which could be a candidate for describing the stationary behaviour of the related Markov fluid queue, but it turns out that the QBD based behaviour is different from the one of the Markov fluid queue.
We examine heterogeneous M/M/c/K server clusters with various load balancing policies. We provide a mathematical framework that allows the efficient computation of the mean system time of jobs in the stationary mean-field limit that allows to find optimal load balancing prioritization.
We consider a stochastic matching model with a general compatibility graph with self-loops on every node and a random matching policy. We consider the discrete time Markov chain associated with such a model where arrivals of items are independently and identically distributed. Due to the self-loops in the compatibility graph, the states of this chain are exactly the independent sets of the graph. We prove that this chain is ordinary lumpable if the automorphism group of the compatibility graph is non-trivial. Additionally, we demonstrate how to construct the partition associated with strong aggregation based on certain subgroups of the automorphism group. This approach can efficiently reduce the size of the state space which could be as large as the exponential of the number of nodes in the compatibility graph before the aggregation. Finally, we illustrate this methodology with examples based on simple compatibility graphs, such as rings and the group of rotations.
Recent advances in AI/ML technologies have accelerated the development of various ML applications. One of the major trends in AI/ML application development is the increasing use of multiple ML models to support high-accuracy inference in a complex end-to-end ML serving. However, testing the right configuration of multiple ML models is expensive, and the application requirements for ML inferences are highly dependent on various factors like the quality of ML models, computing resource performance, and data quality. In this context, techniques and methods that help to emulate and analyze ML inference characteristics using queueing theory can reduce the development effort and cost for ML services encapsulating ML models but also the entire ML system. In this paper, we modeled and analyzed a queueing model for an ML system that uses ensemble learning as an inference method with a new rule and clarified the impacts of model design in ensemble learning on the system's performance. As a result, we demonstrate the usefulness of the analysis for understanding possible configurations and their efficiency in the ML system through queueing analysis and simulation.
We extend the existing theory of formal error bounds for the transient distribution of an aggregated (or lumped) Markov chain when compared to the transient distribution of the original chain, for both discrete- and continuous-time Markov chains. In the discrete-time setting, we bound the stepwise increment of the error, and in the continuous-time setting, we bound the rate at which the error grows. We then compare these error bounds with relevant concepts in the literature such as exact and ordinary lumpability as well as deflatability and aggregatability. These concepts define stricter than necessary conditions to identify settings in which the aggregation error is zero. We also consider possible algorithms for finding suitable aggregations for which the formal error bounds are low, and we analyse first experiments with these algorithms on different models.
